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Full text of "Mathematical And Physical Papers - Iii"

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having the angular velocity of the solid and the initial velocity of the fluid reversed, everything else being the same as before. It follows from symmetry, that at the end of the time t the velocity at P resolved along L will be equal to u', since the motion of the solid and the initial motion of the fluid, which form the data of the one problem, differ from the corresponding quantities in the other problem only as regards the distinction between one way jjjj, ,                              round and the other way round, which has no relation to the
distinction between to and fro in the direction of a line lying :                          in a plane passing through the axis of rotation.    But since all
our equations are linear as regards the velocity, it follows that in the second problem the velocity will be the same as in the first, with a contrary sign, and therefore the velocity at P in the direction of the line L will be equal to — u. Hence u' = — u'} and therefore u'= 0, and therefore the whole motion takes place in annuli about the axis of rotation.
Let the axis of rotation be taken for the axis of z ; let w be ' '                        the angle which a plane passing through this axis and through
the point P makes with the plane of xy, and let v' be the velocity at P. Then
u = — v' sin w,        v = v cos a>,        w = 0,
and all the unknown quantities of the problem are functions of t, z, and Ğr, where tar = \J(x? + y2). Substituting in equations (2) the above values of u, v, and w, and after differentiation putting o> = 0, as we are at liberty to do, we get
/dV    dV     Idv'     v'\      dv'
The first two of these equations give p = a constant, or rather p = a function of t, which for the same reason as in Art. 7 we have a right to suppose to be equal to zero. The third equation combined with the equations of condition serves to determine v.
Now in the particular case of an oscillating disk, the equation (1*79) becomes according to the mode of approximation adopted in Art. 8
dV      <W